In this paper we study the knot Floer homology of a subfamily of twisted (p, q) torus knots where q ≡ ±1 (mod p). Specifically, we classify the knots in this subfamily that admit Lspace surgeries. To do calculations, we use the fact that these knots are (1, 1) knots and, therefore, admit a genus one Heegaard diagram.

Every prism manifold can be parametrized by a pair of relatively prime integers p > 1 and q. In our earlier papers, we determined a complete list of prism manifolds P (p, q) that can be realized by positive integral surgeries on knots in S 3 when q < 0 or q > p; in the present work, we solve the case when 0 < q < p. This completes the solution of the realization problem for prism manifolds.Remark 1.3. If we allow r = −1 in Theorem 1.2, we get p = 2q + 1: see Theorem 1.1.

Abstract. Let P (K) be a satellite knot where the pattern, P , is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, K, is a nontrivial knot in S 3 . We prove that P (K) is an L-space knot if and only if K is an L-space knot and P is sufficiently positively twisted relative to the genus of K. This generalizes the result for cables due to Hedden [Hed09] and the first author [Hom11].

We characterize the (1, 1) knots in the three-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit nontrivial L-space surgeries. We also recover a characterization of the Berge manifold amongst 1-bridge braid exteriors. Introduction.An L-space is a rational homology sphere Y with the "simplest" Heegaard Floer invariant: HF (Y ) is a free abelian group of rank |H 1 (Y ; Z)|. Examples abound and include lens spaces and, more generally, connected sums of manifolds with elliptic geometry [OS05]. One of the most prominent problems in relating Heegaard Floer homology to low-dimensional topology is to give a topological characterization of L-spaces. Work by many researchers has synthesized a bold and intriguing proposal that seeks to do so in terms of taut foliations and orderability of the fundamental group [Juh15, Conjecture 5].A prominent source of L-spaces arises from surgeries along knots. Suppose that K is a knot in a closed three-manifold Y . If K admits a non-trivial surgery to an L-space, then K is an L-space knot. Examples include torus knots and, more generally, Berge knots in S 3 [Ber]; two more constructions especially pertinent to our work appear in [HLV14,Vaf15]. If an L-space knot K admits more than one L-space surgery -for instance, if Y itself is an L-space -then it admits an interval of L-space surgery slopes, so it generates abundant examples of L-spaces [RR15]. With the lack of a compelling guiding conjecture as to which knots are L-space knots, and as a probe of the L-space conjecture mentioned above, it is valuable to catalog which knots in various special families are L-space knots. This is the theme of the present work.The manifolds in which we operate are the rational homology spheres that admit a genus one Heegaard splitting, namely the three-sphere and lens spaces. The knots we consider are the (1, 1) knots in these spaces: these are the knots that can be isotoped to meet each Heegaard solid torus in a properly embedded, boundary-parallel arc. Our main result, Theorem 1.2 below, characterizes (1, 1) L-space knots in simple, diagrammatic terms.A (1, 1) diagram is a doubly-pointed Heegaard diagram (Σ, α, β, z, w), where (Σ, α, β) is a genus one Heegaard diagram of a 3-manifold Y . The (1, 1) knots in Y are precisely those that admit a doubly-pointed Heegaard diagram [GMM05, Hed11, Ras05]. A (1, 1) diagram is reduced if every bigon contains a basepoint. We can transform a given (1, 1) diagram of K into a reduced (1, 1) diagram of K by isotoping the curves into minimal position in the complement of the basepoints: we accomplish this by successively isotoping away bigons in 1 arXiv:1610.04810v2 [math.GT]

We continue our study of the realization problem for prism manifolds. Every prism manifold can be parametrized by a pair of relatively prime integers p > 1 and q. We determine a complete list of prism manifolds P (p, q) that can be realized by positive integral surgeries on knots in S 3 when q > p. The methodology undertaken to obtain the classification is similar to that of the case q < 0 in an earlier paper.